The $K$-closedness property for couples of real Hardy spaces

Mardi 7 avril 2015 14:00-15:00 - Ioann Vasilyev - Université Paris 7 - Denis Diderot

Résumé : It has long been known that in the interpolation sense the scale of analytic Hardy classes on the unit circle behaves in the same way as the scale $L^p$.
For the real method of interpolation, a much stronger rigorous statement is available, specifically the couple $(H^r,H^t)$ is $K$-closed in the couple $(L^r,L^t)$ for all $r,t \in (0,\infty].$
We remind that a subcouple $(F_0,F_1)$ of a couple $(E_0,E_1)$ of quasi-Banach spaces is said to be $K$-closed if any decomposition $f=e_0+e_1, e_i \in E_i$ of a vector $f\in F_0+F_1$ gives rise to a decomposition $f=f_0+f_1$ with $f_i\in F_i$ and $||f_i||_F_i\leq C ||e_i||_E_i, i=1,2.$ When specialized to Hardy classes, this means roughly that any mesurable decomposition of (the boundary function for) an analytic function gives rise to an "analytic’’ decomposition with summands of roughly the same size.
In our talk we will prove the $K$-closedness in the case of the couple of so-called real Hardy spaces $(\mathbb H^r(\mathbb R^n), \mathbb H^p(\mathbb R^n))$ for $\fracn-1n<r<p<\infty$.

Lieu : bât. 425 - 113-115

The $K$-closedness property for couples of real Hardy spaces  Version PDF